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An Axisymmetric Steady State Vortex Ring Model An axisymmetric steady state vortex ring model Ruo-Qian Wanga,∗ aDepartment of Civil and Environmental Engineering, Massachusetts Institute of Technology, MA 02139, USA Abstract Based on the solution of Atanasiu et al. (2004), a theoretical model for ax- isymmetric vortex flows is derived in the present study by solving the vor- ticity transport equation for an inviscid, incompressible fluid in cylindrical coordinates. The model can describe a variety of axisymmetric flows with particular boundary conditions at a moderately high Reynolds number. This paper shows one example: a high Reynolds number laminar vortex ring. The model can represent a family of vortex rings by specifying the modulus func- tion using a Rayleigh distribution function. The characteristics of this vortex ring family are illustrated by numerical methods. For verification, the model results compare well with the recent direct numerical simulations (DNS) in terms of the vorticity distribution and streamline patterns, cross-sectional areas of the vortex core and bubble, and radial vorticity distribution through the vortex center. Most importantly, the asymmetry and elliptical outline of the vorticity profile are well captured. Keywords: vorticity dynamics, Norbury-Fraenkel family, Whittaker function 1. Introduction The studies of vortex rings can be traced back more than one and half centuries, when William Barton Rogers, founder of MIT, conducted the first systematic vortex ring experiments (Rogers, 1858). Inspired by his and other arXiv:1601.06414v1 [physics.flu-dyn] 24 Jan 2016 pioneers' work, a series of theoretical endeavors have been made for the mathematical description of vortex rings. An early example is the famous ∗Corresponding author. Tel.:+1-617-253-6595. Email address: [email protected] (Ruo-Qian Wang) Preprint submitted to Applied Mathematical Modeling June 11, 2021 Hill's vortex, which assumes that vorticity is linearly proportion to radius within a spherical volume, with potential flow outside (Hill, 1894). The assumption is relatively simple, yet it is able to generate realistic-looking streamlines, and is arguably the most popular vortex ring model in applied science and engineering, e.g. Lai et al. (2013). From a different starting point, Fraenkel (1972) analyzed the vortex ring by extending the theoretical solution of a vortex filament to allow for a small finite thickness. To bridge these two models, Norbury (1973) treated the Hill's spherical vortex and Fraenkel's thin ring as two asymptotic members of a series of generalized vortex rings. He then numerically determined a range of intermediate rings, now referred to as the Norbury-Fraenkel (NF) vortex ring family. The NF family delivers more accurate streamlines and more precise vortex ring outlines. However, its linear distribution of vorticity is still unrealistic (Danaila & H´elie,2008). Recently, a solution for viscous vortex rings at low Reynolds numbers (Re) was obtained by Kaplanski & Rudi (1999). They derived a generalized solution to the diffusing viscous vortex ring, by directly solving the axisym- metric vorticity-stream function equations without the nonlinear convection terms. Their solution delineates a donut shape outline and a more realistic Gaussian distribution of vorticity in the radial direction. Kaplanski et al. (2009) extended the solution to turbulence by adopting an effective turbu- lent viscosity. However, recently Danaila & H´elie(2008) conducted Direct Numerical Simulations (DNS) and reported elliptical cross-sections and ra- dial asymmetry for the vorticity distribution, which both Norbury-Fraenkel and Kaplanski-Rudi models are unable to capture. Realizing the issue, Ka- planski et al. (2012) added two adjustable parameters to allow an elliptical cross-section, but the radial asymmetry of the vorticity distribution is still amiss due to their symmetric Gaussian distribution. Laboratory experiments and numerical simulations have also shed light on vortex ring dynamics as reviewed by Lim & Nickels (1995) and Shariff & Leonard (1992). In particular, Gharib et al. (1998) performed experiments in which a piston generated a non-buoyant vortex puff. They observed that if the aspect ratio of piston stroke length L to diameter D was less than about four, the generated vorticity could be incorporated into the head vor- tex, whereas for larger aspect ratios a trailing stem occurs. The phenomenon is now referred to as the \pinch-off", and the critical aspect ratio is called the \formation number". Because the trailing stem stops supplying vorticity to the head vortex after the pinch-off, the \saturated" head vortex ring in the post-formation stage is relatively stable, and should resemble the steady 2 state situation in Hill's vortex. The experimental results can advance the state-of-the-art by enabling a meaningful comparison with idealized theoret- ical models and numerical simulations or experiments with a saturated head vortex ring. An example can be found in Danaila & H´elie(2008), and the present study is also initiated in the same spirit. The best theoretical model would be the analytical solution to the Navier- Stokes equations, which accurately depicts the dynamics of the flow but is difficult to derive. As a compromise, a theoretical model could be built on the solution to the Euler equations, which ignores the viscous effect but captures the more important non-linear dynamics of the flow. In the present case, we focus on the steady state axisymmetric Euler equation, the target of which is to solve an elliptical second order partial differential equation. This equa- tion is a particular form of the Grad-Shafranov equation (Shafranov, 1958; Grad & Rubin, 1958), which is related to the magnetostatic equilibrium in a perfectly conducting fluid and is well known in magnetohydrodynam- ics (MHD). Specifying different forms of the involved arbitrary functions, plasma physicists have derived a series of analytical solutions to this equa- tion, e.g. the simplest solution of the Solov'ev equilibrium (Solov'ev, 1968), the Herrnegger-Maschke solution (Herrnegger, 1972; Maschke, 1973), and the recent more general solution by Atanasiu et al. (2004) (hereafter referred to as AGLM). It can be shown that the Hill's spherical vortex corresponds to the Solov'ev equilibrium, yet no counterparts of the more advanced solutions have been explored for vortex dynamics. This encourages us to take advan- tage of them, especially the AGLM, to reach a more accurate model of vortex rings. Beyond that, a higher accuracy model can also meet the industrial de- mand to improve the description of vortex structures to replace the aged Hill's vortex, e.g. Lai et al. (2013). In particular, a more realistic vortic- ity distribution may lay a better foundation to address the particle-vorticity interactions, which motivates us to make the following study. As mentioned earlier, existing models have improved progressively to de- scribe more detailed features of real vortex rings. However, an outstanding issue is the asymmetry of the vorticity distribution. The present paper pro- vides an alternative analysis that is capable of incorporating the asymmetry, by deriving a theoretical model for high Re laminar vortex rings based on the solution to the axisymmetric inviscid vortex flow. The governing equations are first presented in section 2 and shown to relate to the Grad-Shafranov equation. The alternative vortex ring solution is shown in section 3, and its properties are derived in section 4. Then, the model is compared to existing 3 simulation results in section 5. Summary and conclusions are given in section 6. 2. A particular solution We adopt the cylindrical coordinate system (r; φ, x), where x is the local longitudinal coordinate. The incompressible axisymmetric vorticity trans- port equations can be stated as (Fukumoto & Kaplanski, 2008) @ξ @ @ µ @2ξ @2ξ 1 @ξ ξ + (vξ) + (uξ) = + + − (1) @t @r @x ρ @x2 @r2 r @r r2 where t is time, ξ is the vorticity, u and v are velocity components in x and r directions respectively, µ is the dynamic viscosity, and ρ is the density. Introducing the Stokes stream function , the velocity components can then be expressed as 1 @ 1 @ u = − + U ; v = ; (2) r @r x r @x where Ux is the translational velocity of the vortex centroid. The vorticity can be defined by the stream function as @2 @2 1 @ + − = −rξ; (3) @r2 @x2 r @r which provides the closure to solve (1). Equations (1)-(3) can therefore be applied to any axisymmetric flows in theory. Substituting (2) into (1) and normalizing the variables by a length scale l and a velocity scale U, i.e. ∗ ∗ ∗ ∗ 2 ∗ −1 ∗ u = Uu ; v = Uv ; r = lr ; x − Uxt = lx ; = l U ; ξ = Ul ξ ; (4) a non-dimensionalized form of (1) can be derived as @ξ∗ @ ξ∗ @ @ ξ∗ @ @ − U + − + (U ξ∗) x @x∗ @r∗ r∗ @x∗ @x∗ r∗ @r∗ @x∗ x 1 @2ξ∗ @2ξ∗ 1 @ξ∗ ξ∗ = + + − ; (5) Re @x∗2 @r∗2 r∗ @r∗ r∗2 4 where Re = ρUl/µ. U and l can be specified for a particular application. Note that if Re 1, the nonlinear terms on the LHS of (5) can be neglected, leading to the theoretical solution obtained earlier by Kaplanski & Rudi (1999). For Re 1 but before transition into turbulence, the viscous effect (RHS of equation (5)) can be ignored in the high Re laminar flows. Assuming a steady translation such that Ux is constant, and dropping the * sign for brevity, then @ ξ @ @ ξ @ − = 0: (6) @r r @x @x r @r Using ξ = rf( ) (Shariff & Leonard, 1992) and substituting into (3), we get @2 @2 1 @ + − = −r2f ( ) : (7) @r2 @x2 r @r As mentioned earlier, equation (7) is a particular form of the Grad-Shafranov equation: @2 @2 1 @ + − = −r2f ( ) + g ( ) ; (8) @r2 @x2 r @r with the arbitrary function g( ) = 0. The simplest particular solution to (8) is the Solov'ev equilibrium solu- tion, which assumes f( ) = A; g( ) = B (9) where A and B are constants.
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